\label{prop:chain-verification-protocol}
The following protocol verifies the complete dependency chain
(Definition~\ref{def:complete-dependency-chain})
at primorial depth~$k$:
\begin{enumerate}
\item \textbf{Link audit.}
      For each link~$n$ ($1 \le n \le 14$),
      identify the earning theorem or definition
      and confirm it appears in the stated chapter
      with $\tau$-effective status.

\item \textbf{Primorial instantiation.}
      Replace $\T$ by~$\T_k$,
      $\Lemniscate$ by~$\Lemniscate_k$,
      $\iota_\tau$ by~$\iota_{\tau,k}$.
      Verify each link by explicit computation
      in $\mathbb{Z}/\operatorname{Prim}(k)\mathbb{Z}$.

\item \textbf{Compatibility.}
      For adjacent levels $(k, k{+}1)$,
      verify that the $(k{+}1)$-chain restricts
      to the $k$-chain under
      $\mathbb{Z}/\operatorname{Prim}(k{+}1)\mathbb{Z}
       \twoheadrightarrow
       \mathbb{Z}/\operatorname{Prim}(k)\mathbb{Z}$.

\item \textbf{Inverse-limit assembly.}
      Verify the Mittag-Leffler condition so that
      $\varprojlim_k (\text{chain at level } k)$
      recovers the full chain.

\item \textbf{Scope confirmation.}
      Confirm every link carries $\tau$-effective scope.
\end{enumerate}
A chain passing all five checks at every $k \ge 1$
is \emph{verified}.